Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 472721, 8 pages
doi:10.1155/2010/472721
Research Article
Approximate Behavior of Bi-Quadratic Mappings
in Quasinormed Spaces
Won-Gil Park1 and Jae-Hyeong Bae2
1
Department of Mathematics Education, College of Education, Mokwon University,
Daejeon 302-729, Republic of Korea
2
College of Liberal Arts, Kyung Hee University, Yongin 446-701, Republic of Korea
Correspondence should be addressed to Jae-Hyeong Bae, jhbae@khu.ac.kr
Received 27 April 2010; Accepted 18 June 2010
Academic Editor: Radu Precup
Copyright q 2010 W.-G. Park and J.-H. Bae. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We obtain the generalized Hyers-Ulam stability of the bi-quadratic functional equation fx y, z w fx y, z − w fx − y, z w fx − y, z − w 4fx, z fx, w fy, z fy, w in
quasinormed spaces.
1. Introduction
In 1940, Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin
in which he discussed a number of unsolved problems, containing the stability problem of
homomorphisms as follows
Let G1 be a group and let G2 be a metric group with the metric d·, ·. Given ε > 0, does there
exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality dhxy, hxhy < δ for
all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with dhx, Hx < ε for all x ∈ G1 ?
Hyers 2 proved the stability problem of additive mappings in Banach spaces. Rassias
3 provided a generalization of Hyers theorem which allows the Cauchy difference to be
unbounded: let f : E → E be a mapping from a normed vector space E into a Banach space
E subject to the inequality
f x y − fx − f y ≤ xp xp
1.1
for all x, y ∈ E, where and p are constants with > 0 and p < 1. The above inequality
provided a lot of influence in the development of a generalization of the Hyers-Ulam stability
2
Journal of Inequalities and Applications
concept. Găvruţa 4 provided a further generalization of Hyers-Ulam theorem. A square
norm on an inner product space satisfies the important parallelogram equality:
x y2 x − y2 2x2 2y2 .
1.2
f x y f x − y 2fx 2f y
1.3
The functional equation
is called the quadratic functional equation whose solution is said to be a quadratic mapping.
A generalized stability problem for the quadratic functional equation was proved by Skof 5
for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa
6 noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by
an Abelian group. Czerwik 7 proved the generalized stability of the quadratic functional
equation, and Park 8 proved the generalized stability of the quadratic functional equation
in Banach modules over a C∗ -algebra.
Throughout this paper, let X and Y be vector spaces.
Definition 1.1. A mapping f : X × X → Y is called bi-quadratic if f satisfies the system of the
following equations:
f x y, z f x − y, z 2fx,
z 2f y, z ,
f x, y z f x, y − z 2f x, y 2fx, z.
1.4
When X Y R, the function f : R × R → R given by fx, y : ax2 y2 is a solution
of 1.4.
For a mapping f : X × X → Y , consider the functional equation:
f x y, z w f x y, z − w f x − y, z w f x − y, z − w
4 fx, z fx, w f y, z f y, w .
1.5
Definition 1.2 see 9, 10. Let X be a real linear space. A quasinorm is real-valued function
on X satisfying the following
i x ≥ 0 for all x ∈ X and x 0 if and only if x 0.
ii λx |λ|x for all λ ∈ R and all x ∈ X.
iii There is a constant K ≥ 1 such that x y ≤ Kx y for all x, y ∈ X.
It follows from the condition iii that
2m 2m
xi ≤ K m xi ,
i1 i1
for all m ≥ 1 and all x1 , x2 , . . . , x2m1 ∈ X.
2m1
2m1
xi ≤ K m1
xi i1 i1
1.6
Journal of Inequalities and Applications
3
The pair X, · is called a quasinormed space if · is a quasinorm on X. The smallest
possible K is called the modulus of concavity of · . A quasi-Banach space is a complete
quasinormed space. A quasinorm · is called a p-norm 0 < p ≤ 1 if
x yp ≤ xp yp
1.7
for all x, y ∈ X. In this case, a quasi-Banach space is called a p-Banach space.
Given a p-norm, the formula dx, y : x − yp gives us a translation invariant
metric on X. By the Aoki-Rolewicz theorem 10 see also 9, each quasinorm is equivalent
to some p-norm. Since it is much easier to work with p-norms, henceforth we restrict our
attention mainly to p-norms. In 11, Tabor has investigated a version of Hyers-Rassias-Gajda
theorem see also 3, 12 in quasi-Banach spaces. Since then, the stability problems have been
investigated by many authors see 13–18.
The authors 19 solved the solutions of 1.4 and 1.5 as follows.
Theorem A. A mapping f : X × X → Y satisfies 1.4 if and only if there exist a multi-additive
mapping M : X × X × X × X → Y such that fx, y Mx, x, y, y and Mx, y, z, w My, x, z, w Mx, y, w, z for all x, y, z, w ∈ X.
Theorem B. A mapping f : X × X → Y satisfies 1.4 if and only if it satisfies 1.5.
In this paper, we investigate the generalized Hyers-Ulam stability of 1.4 and 1.5 in
quasi-Banach spaces.
2. Stability of 1.4 and 1.5 in Quasi-normed Spaces
Throughout this section, assume that X is a quasinormed space with quasinorm · X and
that Y is a p-Banach space with p-norm · Y . Let K be the modulus of concavity of · Y .
Let ϕ : X × X × X → 0, ∞ and ψ : X × X × X → 0, ∞ be two functions such that
1 n
ϕ 2 x, 2n y, z 0,
n
n→∞4
lim
1 n
ψ 2 x, y, z 0,
n
n→∞4
lim
1 ϕ x, y, 2n z 0,
n
n→∞4
lim
1 lim ψ x, 2n y, 2n z 0
n → ∞ 4n
2.1
for all x, y, z ∈ X.
Let ϕ, ψ : X × X × X → 0, ∞ be two functions satisfying
for all x, y, z ∈ X.
∞
p
1 j
M x, y, z :
ϕ 2 x, 2j y, z < ∞,
pj
j0 4
2.2
∞
p
1 j
j
N x, y, z :
ψ
x,
2
y,
2
z
<∞
pj
j0 4
2.3
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Journal of Inequalities and Applications
Theorem 2.1. Let f : X × X → Y be a mapping such that
fx y, z fx − y, z − 2fx, z − 2fy, z ≤ ϕ x, y, z ,
Y
fx, y z fx, y − z − 2fx, y − 2fx, z ≤ ψ x, y, z ,
Y
2.4
2.5
and let fx, 0 0 and f0, y 0 for all x, y, z ∈ X. Then there exist two bi-quadratic mappings
F1 , F2 : X × X → Y such that
fx, y − F1 x, y ≤ 1 M x, x, y 1/p ,
Y
4
fx, y − F2 x, y ≤ 1 N x, y, y 1/p
Y
4
2.6
2.7
for all x, y ∈ X.
Proof. Letting y x in 2.4, we get
fx, z − 1 f2x, z ≤ 1 ϕx, x, z
4
4
Y
2.8
for all x, z ∈ X. Thus we have
1
f2j x, z − 1 f2j1 x, z ≤ 1 ϕ 2j x, 2j x, z
4j
4j1
4j1
Y
2.9
for all x, z ∈ X. Replacing z by y in the above inequality, we obtain
1
f2j x, y − 1 f 2j1 x, y ≤ 1 ϕ 2j x, 2j x, y
4j
4j1
4j1
Y
2.10
for all x, y ∈ X. Since Y is a p-Banach space, for given integers l, m 0 ≤ l < m, we see that
p m−1
1
p
1 j
f2l x, y − 1 f2m x, y ≤
f 2 x, y − 1 f 2j1 x, y 4l
j
4m
4j1
Y
Y
jl 4
p
1 1 m−1
≤ p
ϕ 2j x, 2j x, y
pj
4 jl 4
2.11
for all x, y ∈ X. By 2.2 and 2.11, the sequence {1/4j f2j x, y} is a Cauchy sequence for
all x, y ∈ X. Since Y is complete, the sequence {1/4j f2j x, y} converges for all x, y ∈ X.
Define F1 : X × X → Y by
1 F1 x, y : lim j f 2j x, y
j →∞4
2.12
Journal of Inequalities and Applications
5
for all x, y ∈ X. Putting l 0 and taking m → ∞ in 2.11, one can obtain the inequality 2.6.
By 2.4 and 2.5, we get
1 f x y, 2j z 1 f x − y, 2j z − 2 1 f x, 2j z − 2 1 f y, 2j z ≤ 1 ϕ x, y, 2j z ,
4j
4j
4j
4j
4j
Y
1
f2j x, y z 1 f2j x, y − z − 2 1 f 2j x, y − 2 1 f 2j x, z ≤ 1 ψ 2j x, y, z
4j
4j
4j
4j
4j
Y
2.13
for all x, y, z ∈ X and all j. Letting j → ∞ in the above two inequalities and using 2.1, F1 is
bi-quadratic.
Next, setting z y in 2.5,
fx, y − 1 fx, 2y ≤ 1 ψ x, y, y
4
4
Y
2.14
for all x, y ∈ X. By the same method as above, define F2 : X × X → Y by F2 x, y :
limj → ∞ 1/4j fx, 2j y for all x, y ∈ X. By the same argument as above, F2 is a bi-quadratic
mapping satisfying 2.7.
Corollary 2.2. Let f : X × X → Y be a mapping such that
fx y, z fx − y, z − 2fx, z − 2fy, z ≤ δ,
Y
fx, y z fx, y − z − 2fx, y − 2fx, z ≤ ε,
Y
2.15
and let fx, 0 0 and f0, y 0 for all x, y, z ∈ X. Then there exist two bi-quadratic mappings
F1 , F2 : X × X → Y such that
fx, y − F1 x, y ≤ √ δ
,
Y
p
4p − 1
fx, y − F2 x, y ≤ √ ε
Y
p
4p − 1
2.16
for all x, y ∈ X.
Proof. In Theorem 2.1, putting ϕx, y, z : δ and ψx, y, z : ε for all x, y, z ∈ X, we get the
desired result.
From now on, let ϕ : X × X × X × X → 0, ∞ be a function such that
lim
1
n → ∞ 16n
ϕ 2n x, 2n y, 2n z, 2n w 0,
∞
p
1 j
L x, y, z, w :
ϕ 2 x, 2j y, 2j z, 2j w < ∞
pj
j0 16
for all x, y, z, w ∈ X.
2.17
2.18
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Journal of Inequalities and Applications
We will use the following lemma in order to prove Theorem 2.4.
Lemma 2.3 see 20. Let 0 < p ≤ 1 and let x1 , x2 , . . . , xn be nonnegative real numbers. Then
⎞p
⎛
n
n
⎝ xj ⎠ ≤
xj p .
j1
2.19
j1
Theorem 2.4. Let f : X × X → Y be a mapping such that
f x y, z w f x y, z − w f x − y, z w f x − y, z − w
−4fx, z − fx, w − fy, z − fy, wY ≤ ϕ x, y, z, w ,
2.20
and let fx, 0 0 and f0, y 0 for all x, y, z, w ∈ X. Then there exists a unique bi-quadratic
mapping F : X × X → Y such that
fx, y − F x, y ≤ 1 L x, x, y, y 1/p
Y
16
2.21
for all x, y ∈ X.
Proof. Letting y x and w z in 2.20, we have
fx, z − 1 f2x, 2z ≤ 1 ϕx, x, z, z
16
16
Y
2.22
for all x, z ∈ X. Thus we obtain
1
1
1
j
j
j1
j1 j
j
j
j
16j f2 x, 2 z − 16j1 f2 x, 2 z ≤ 16j1 ϕ 2 x, 2 x, 2 z, 2 z
Y
2.23
for all x, z ∈ X and all j. Replacing z by y in the above inequality, we see that
1
1
j
j
j1
j1
≤ 1 ϕ 2j x, 2j x, 2j y, 2j y
f2
x,
2
y
−
f2
x,
2
y
16j
j1
j1
16
16
Y
2.24
for all x, y ∈ X and all j. By Lemma 2.3, for given integers l, m 0 ≤ l < m, we get
p m−1
p
1
1 j
1
j
j1
j1
f2l x, 2l y − 1 f2m x, 2m y ≤
f
2
x,
2
y
−
f
2
x,
2
y
16l
j
16m
16j1
Y
Y
jl 16
p
1 1 m−1
j
j
j
j
≤
ϕ
2
x,
2
x,
2
y,
2
y
16 jl 16pj
2.25
Journal of Inequalities and Applications
7
for all x, y ∈ X. By 2.18 and 2.25, the sequence {1/16j f2j x, 2j y} is a Cauchy sequence
for all x, y ∈ X. Since Y is complete, the sequence {1/16j f2j x, 2j y} converges for all
x, y ∈ X. Define F : X × X → Y by
1 F x, y : lim j f 2j x, 2j y
j → ∞ 16
2.26
for all x, y ∈ X.
By 2.20, we have
1 j
j
j
1 j
f
2
f
2
w
−
w
x
y
,
2
x
y
,
2
z
z
16j
16j
1 1 j f 2j x − y , 2j z w j f 2j x − y , 2j z − w
16
16
4
4
4 4 − j f 2j x, 2j z − j f 2j x, 2j w − j f 2j y, 2j z − j f 2j y, 2j w 16
16
16
16
Y
1
≤ j ϕ 2j x, 2j y, 2j z, 2j w
16
2.27
for all x, y, z, w ∈ X and all j. Letting j → ∞ and using 2.17, we see that F satisfies 1.5. By
Theorem B, we obtain that F is bi-quadratic. Setting l 0 and taking m → ∞ in 2.25, one
can obtain the inequality 2.21. If G : X × X → Y is another bi-quadratic mapping satisfying
2.21, we obtain
Fx, y − Gx, yp
Y
1 F2n x, 2n y − G2n x, 2n yp
Y
pn
16
p
p
1 1 ≤ pn F2n x, 2n y − f2n x, 2n yY pn f2n x, 2n y − G2n x, 2n yY
16
16
≤
1 1 n
L 2 x, 2n x, 2n y, 2n y −→ 0
pn
8 16
2.28
as n −→ ∞
for all x, y ∈ X. Hence the mapping F is the unique bi-quadratic mapping, as desired.
Corollary 2.5. Let ε be a nonnegative real number. Let f : X × X → Y be a mapping such that
f x y, z w f x y, z − w f x − y, z w f x − y, z − w
−4 fx, z − fx, w − f y, z − f y, w Y ≤ ε,
2.29
8
Journal of Inequalities and Applications
and let fx, 0 0 and f0, y 0 for all x, y, z, w ∈ X. Then there exists a unique bi-quadratic
mapping F : X × X → Y such that
fx, y − Fx, y ≤ √ ε
Y
p
16p − 1
2.30
for all x, y ∈ X.
Proof. In Theorem 2.4, putting ϕx, y, z, w : ε for all x, y, z, w ∈ X, we get the desired result.
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